Yury Volkov (wolf86 666@list.ru).Abstract. The work is devoted to the variety of 2-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of certain algebra series in the variety of 2-dimensional algebras. Finally, we apply our results to obtain analogous descriptions for the subvarieties of flexible, and bicommutative algebras. In particular, we describe rigid algebras and irreducible components for these subvarieties.

), Alexandre Pozhidaev (app@math.nsc.ru), Yury Volkov (wolf86 666@list.ru).Abstract. We describe degenerations of four-dimensional Zinbiel and four-dimensional nilpotent Leibniz algebras over C. In particular, we describe all irreducible components in the corresponding varieties.

We give a complete algebraic classification of nilpotent binary Lie algebras of dimension at most 6 over an arbitrary base field of characteristic not 2 and a complete geometric classification of nilpotent binary Lie algebras of dimension 6 over C. As an application, we give an algebraic and geometric classification of nilpotent anticommutative CD-algebras of dimension at most 6.Proof. Let θ ∈ Z 2 BL (A, F). Then A θ is a binary Lie algebra. We denote the Jacobian of elements x, y, z in A θ by J A θ (x, y, z). Now consider x, y, z, t ∈ A. (1.3), By the identityt] , x, y) = 0; J([x, y] , z, t) + J([x, t] , z, y) + J([z, y] , x, t) + J([z, t] , x, y) = 0; we deduce that ψ θ ([x, y] , z, t) + ψ θ ([x, t] , z, y) + ψ θ ([z, y] , x, t) + ψ θ ([z, t] , x, y) = 0, as desired.

Yury Volkov (wolf86 666@list.ru).Abstract. We describe degenerations of four-dimensional binary Lie algebras, and five-and six-dimensional nilpotent Malcev algebras over C. In particular, we describe all irreducible components of these varieties.

This paper is devoted to give the complete algebraic and geometric classification of 4dimensional nilpotent Novikov algebras over C.